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Chapter 2: Problem 6

Find the best-fitting straight line to the given set of data, using the method of least squares. Graph this straight line on a scatter diagram. Find the correlation coefficient. $$ (0,0),(1,1),(2,2),(3,4) $$

Short Answer

Expert verified
The best-fit line is \(y = 0.9x + 0.4\) with a correlation coefficient of 0.894.

Step by step solution

01

Define the Equation of the Line

The equation of the line we want to fit to the data is given by the formula \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
02

Use the Least Squares Formula for Slope

The slope \(m\) of the best-fit line can be calculated using the formula: \[ m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2} \] where \(n\) is the number of points, \(\Sigma xy\) is the sum of the products of each pair \((x,y)\), \(\Sigma x\) and \(\Sigma y\) are the sums of all \(x\) and \(y\) values, respectively, and \(\Sigma x^2\) is the sum of the squares of the \(x\) values.
03

Calculate Sums for Formula Substitution

Calculate and substitute the values into the formula: \(n = 4\), \(\Sigma x = 6\), \(\Sigma y = 7\), \(\Sigma xy = 15\), \(\Sigma x^2 = 14\).
04

Solve for the Slope \(m\)

Substitute the calculated sums into the formula for \(m\): \[ m = \frac{4(15) - 6 \times 7}{4(14) - 6^2} = \frac{60 - 42}{56 - 36} = \frac{18}{20} = 0.9 \]
05

Use the Least Squares Formula for Intercept

The y-intercept \(c\) can be calculated using: \[ c = \frac{\Sigma y - m \Sigma x}{n} \] Substitute the known values: \[ c = \frac{7 - 0.9 \times 6}{4} = \frac{7 - 5.4}{4} = \frac{1.6}{4} = 0.4 \]
06

Formulate the Equation of Best-Fit Line

Substitute the calculated \(m\) and \(c\) back into the line equation: \(y = 0.9x + 0.4\).
07

Calculate the Correlation Coefficient

The correlation coefficient \(r\) is calculated using: \[ r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n(\Sigma x^2) - (\Sigma x)^2][n(\Sigma y^2) - (\Sigma y)^2]}} \] Substituting known values and calculating \(\Sigma y^2 = 21\), we find: \[ r = \frac{4(15) - 6 \times 7}{\sqrt{[4(14) - 6^2][4(21) - 7^2]}} = \frac{18}{\sqrt{80}} = 0.894 \]
08

Graph the Line on Scatter Plot

Plot the points \((0,0), (1,1), (2,2), (3,4)\) on the graph. Draw the line \(y = 0.9x + 0.4\) to see how it fits the data points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

best-fit line
The best-fit line is an essential tool in statistics for analyzing the relationship between two variables. When we talk about finding the best-fit line, we often refer to the line that best represents the trend shown by a set of data points on a graph. This is typically done using the least squares method, which minimizes the sum of the squares of the differences between observed and estimated values.

To determine the best-fit line, we use the linear equation format:
  • The formula is given by \( y = mx + c \), where \( m \) is the slope, and \( c \) is the y-intercept.
  • The slope \( m \) represents how much \( y \) changes for a unit change in \( x \). A positive slope means a positive relationship, while a negative slope indicates a negative one.
  • The y-intercept \( c \) is the value of \( y \) when \( x = 0 \).
In our example, we calculated the slope \( m = 0.9 \) and the intercept \( c = 0.4 \), leading to the equation of the line: \( y = 0.9x + 0.4 \). This line attempts to be as close as possible to all of the data points given.
scatter diagram
A scatter diagram, also known as a scatter plot, is a two-dimensional graph used to visualize the relationship between two variables. The values of one variable are plotted along the x-axis, while the values of the other are plotted along the y-axis. Scatter diagrams are incredibly helpful and provide a visual representation of the data, which can reveal trends, patterns, and potential correlations between the variables.

When analyzing the data
  • Each point plotted on the scatter diagram represents an observation from your dataset.
  • Observing the scatter of points can show how strongly two variables are related.
  • The best-fit line, if added, helps in seeing the overall trend.
For our specific data points
  • The points \( (0,0), (1,1), (2,2), (3,4) \) can be plotted on the scatter diagram.
  • Drawing the line \( y = 0.9x + 0.4 \) on the scatter plot provides a visual display of the relationship between the x and y values.
correlation coefficient
The correlation coefficient, represented by the letter \( r \), quantifies the strength and direction of a linear relationship between two variables. It is a crucial statistical tool used to understand how closely data points align with a best-fit line.

The correlation coefficient has specific properties:
  • The value of \( r \) ranges from \(-1\) to \(1\).
  • A value close to \(1\) indicates a strong positive linear relationship, while a value close to \(-1\) suggests a strong negative linear relationship.
  • If \( r \) is around \(0\), there is little to no linear relationship between the variables.
In the given exercise, the calculated correlation coefficient is \( r = 0.894 \), which suggests a strong positive relationship between the x and y variables. This means that as \( x \) increases, \( y \) tends to increase as well, fitting well with the trend depicted by the data points and the constructed best-fit line.

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Most popular questions from this chapter

Moose Reproductive Effort Ericsson and colleagues \(^{31}\) studied the effect of the age of a female moose on the mortality of her offspring. They collected data shown in the table relating the age of the female moose to offspring mortality during the hunting season. $$ \begin{array}{|l|cccc|} \hline \text { Moose age } & 2 & 3 & 4 & 5 \\\ \hline \text { Mortality of Offspring } & 0.5 & 0.4 & 0.25 & 0.35 \\ \hline \text { Moose age } & 6 & 7 & 8 & 9 \\ \hline \text { Mortality of Offspring } & 0.35 & 0.5 & 0.37 & 0.35 \\ \hline \text { Moose age } & 10 & 11 & 12 & 13 \\\ \hline \text { Mortality of Offspring } & 0.48 & 0.37 & 0.53 & 0.38 \\\ \hline \text { Moose age } & 14 & & & \\ \hline \text { Mortality of Offspring } & 0.60 & & & \\ \hline \end{array} $$ a. Find the best-fitting quadratic (as the researches did) relating age to mortality of offspring and the square of the correlation coefficient. b. Find the age at which mortality of offspring is minimized.

Ecological Entomology Elliott \(^{37}\) studied the temperature effects on the alder fly. In 1967 he collected the data shown in the following table relating the temperature in degrees Celsius to the number of pupae successfully completing pupation. $$ \begin{array}{|c|cccccc|} \hline t & 8 & 10 & 12 & 16 & 20 & 22 \\ \hline y & 18 & 27 & 43 & 44 & 37 & 5 \\ \hline \end{array} $$ a. Use quadratic regression to find \(y\) as a function of \(t\). b. Determine the temperature at which this model predicts the maximum number of successful pupations. c. Determine the two temperatures at which this model predicts there will be no successful pupation.

Nordin \(^{49}\) obtained the following data relating output and total final cost for an electric utility in Iowa. (Instead of using the original data set for 541 eight-hour shifts, we have just given the array means.) $$\begin{array}{|l|llllllll}\hline x & 25 & 28 & 33 & 38 & 41 & 45 & 50 & 53 \\\\\hline y & 24 & 24 & 26 & 28 & 27 & 31 & 34 & 38 \\\\\hline x & 57 & 63 & 66 & 71 & 74 & 79 & 83 & 89 \\\\\hline y & 39 & 42 & 43 & 51 & 53 & 52 & 54 & 52 \\\\\hline\end{array} $$ Here \(x\) is the output as a percent of capacity, and \(y\) is final total cost in dollars (multiplied by a constant not given to avoid showing the exact level of the costs.) Use cubic regression to find the best-fitting cubic to the data and the correlation coefficient. Graph. Also find the correlation coefficients associated with both linear and quadratic regression, and compare them to that found for cubic regression.

VCR Sales The following table gives the sales of VCRs in the United States in millions for some selected years. \(^{72}\) a. On the basis of the data given for the years 1978 to 1988 , find the best- fitting exponential function using exponential regression. Determine the correlation coefficient. Graph. Using this model, estimate sales in \(1992 .\) b. Now find the best-fitting logistic curve. Graph. Using this model, estimate sales in \(1992 .\) Note that the actual factory sales in 1992 was 66.78 million. $$ \begin{array}{|l|cccccc|} \hline \text { Year } & 1978 & 1980 & 1982 & 1984 & 1986 & 1988 \\ \hline \text { Sales } & 0.20 & 0.84 & 2.53 & 8.88 & 30.92 & 51.39 \\ \hline \end{array} $$

Productivity Bernstein \(^{14}\) also studied the correlation between investment as a percent of GNP and productivity growth of six countries: France (F), Germany (G), Italy (I), Japan (J), the United Kingdom (UK), and the United States (US). Productivity is given as output per employeehour in manufacturing. The data they collected for the years \(1960-1977\) is given in the following table. \begin{tabular}{|c|cccccc|} \hline Country & US & UK & I & F & G & J \\ \hline\(x\) & 17 & 18 & 22 & 23 & 24 & 34 \\ \hline\(y\) & 2.8 & 3.0 & 5.6 & 5.5 & 5.8 & 8.3 \\ \hline \end{tabular} Here \(x\) is investment as percent of GNP, and \(y\) is the productivity growth (\%). a. Determine the best-fitting line using least squares and the correlation coefficient. b. What does this model predict the productivity growth will be when investment is \(20 \%\) of GNP? c. What does this model predict investment as a percent of GNP will be if productivity growth is \(7 \% ?\)
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